Investigation of stochastic inventory models of type (s, S) with dependent components

The study is concerned with a stochastic inventory model of the type (s,S) with dependent components. Two various modifications of the renewal-reward process (X(t)) are constructed mathematically for the purpose of analysing this model. It is notable that, due to their complex mathematical structure, stochastic processes with dependent components have only been adressed to a limited extent in the literature. This gap is partially addressed by the present study. The effect of the dependency condition on the stationary characteristics of X(t) is investigated. For this, in the first chapter of the study, the renewal-reward process with dependent components is constructed without any discrete interference of chance. First of all, it is shown that the process is ergodic under some weak conditions. Subsequently, the precise result for the limit distribution of X(t) is acquired. Subsequently, assuming linear dependence, the asymptotic result for the limit distribution of X(t) is provided. Furthermore, it is proved that as β≡S-s→∞, the limiting distribution of V(t) weakly converges to the uniform distribution on the interval (0,1), which defines the standardised process V(t)≡X(t)/β ∈[0,1]. This result allows to conclude that the limit distribution of X(t) is approximately close to a uniform distribution in the interval [0,β]. Furthermore, this study provides precise results for the steady-state characteristics of the limit disribution of X(t). Subsequently, the asymptotic behaviour of the stationary characteristics of the limit distribution is analysed by taking advantage of the fundamental results of renewal theory and the Laplace transform. Asymptotic expansions for the stationary characteristics of the limit distribution are revealed based on the analysis. Additionally, asymptotic results are proposed for the expected value, variance and standard deviation of X(t). Eventually, the asymptotic results presented for the stationary characteristics of the limit distribution of X(t) are analysed by using special examples under linear and nonlinear dependence assumptions. The second chapter of the paper presents a detailed examination and analysis of the renewal-reward process, taking into account dependency assumption and discrete interference of chance. In this study, the discrete interference of chance is represented by a truncated normal distribution within the interval [0,β]. The proof of ergodicity and the precise result for the limit distribution are presented. Furthermore, an precise formula for the stationary characteristics of the limit distribution of X(t) is also attained using this result. Nevertheless, the approximate formulas of the steady-state characteristics of the limit distribution of X(t) has been investigated, resulting in the derivation of approximate formulas due to the inherent mathematical challenges associated with the precise formula. To illustrate the results, specific examples are presented.